So the issue is that binary is a very efficient way of communicating a number between 1 and 64, but less efficient at communicating a number between say say 1 and 49. However using 7-ary digits is efficient for communicating a number between 1 and 49. It was noted several times in this blog that the solution to the original problem can be expressed in terms of Hamming codes, and thus works for q-ary digits for any prime power q.

In particular, given |P^2(F_7)|=8 7-ary digits, by changing at most one you can communicate a number between 1 and 49.

From the original problem we know that given 7 bits, you can communicate any 7-ary digit by flipping at most one bit. Thus replace each 7-ary digit with 7 bits to get 7*8=56 bits and you can communicate a number between 1 and 49 by flipping at most one bit.

Finally add a dud bit, to change if you do not need to flip anything, and we have a_{57} is at least 49, disproving the conjecture.

]]>By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number

.

By perturbations of problem A by means problem B

,

where is a non-decreasing -order-convex function on a partially set and .

Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .

Theorem. Let and be guaranteed error estimates for the gradient algorithm in Problems A and B, respectively. Then .

To prove Theorem, we need the following lemma.

Lemma. The gradient maximum and the global maximum of any -ordered-convex non-decreasing function on are connected by the following relations:

, (1)

where

– is the set of all maximal elements of the partially ordered set .

Proof of Lemma. By virtue of item of Theorem 4 [4], we have for

Together with the fact that

the last inequality yields

.

Therefore

,

Where

Then, by repeating the scheme of the proof of Theorem 4 [4], we obtain estimates (1). Lemma is proved.

Proof of Theorem. According to LemmaFinally we should note some rather confusing facts. Although given Jordan’s work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation Ax= b is not. The Jordan of Gauss-Jordan is Wilhelm Jordan (1842 to 1899) who applied the method to finding squared errors to work on surveying. Jordan algebras are called after the German physicist and mathematician Pascual Jordan (1902 to 1980).

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Pythagora

• ms on the

• limit of a function as x approaches a fixed constant

• limit of a function as x approaches plus or minus infinity

• limit of a function using the precise epsilon/delta definition of limit

• limit of a function using l’Hopital’s rule

,

,

where – is a non-decreasing -order-convex function on a partially set .

Let be an optimal solution of Problem A, and let be the point obtained by the following iterative procedure [4]:

which halts on the step if either or is the maximal element of the set (the set contains the zero , as we have stipulated). This point is called the gradient maximum os the function on the set [4].

By a guaranteed error estimate for the gradient algorithm in Problem A we mean a number

.

By perturbations of problem A by means problem B

,

where is a non-decreasing -order-convex function on a partially set and .

Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .

Theorem. Let and be guaranteed error estimates for the gradient algorithm in Problems A and B, respectively. Then .

To prove Theorem, we need the following lemma.

Lemma. The gradient maximum and the global maximum of any -ordered-convex non-decreasing function on are connected by the following relations:

, (1)

where

– is the set of all maximal elements of the partially ordered set .

Proof of Lemma. By virtue of item of Theorem 4 [4], we have for

Together with the fact that

the last inequality yields

.

Therefore

,

Where

]]>Thank you! I have still been wondering.

]]>The centre-left may have lost in the battle of ideas in the 1980s, but we are winning now. And we have won a bigger battle today: the battle of values. The challenge we face has to be met by us together: one nation, one community.

When a young ]]>